Best Known (51−18, 51, s)-Nets in Base 4
(51−18, 51, 130)-Net over F4 — Constructive and digital
Digital (33, 51, 130)-net over F4, using
- 3 times m-reduction [i] based on digital (33, 54, 130)-net over F4, using
- trace code for nets [i] based on digital (6, 27, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 27, 65)-net over F16, using
(51−18, 51, 162)-Net over F4 — Digital
Digital (33, 51, 162)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(451, 162, F4, 18) (dual of [162, 111, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(451, 255, F4, 18) (dual of [255, 204, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(451, 255, F4, 18) (dual of [255, 204, 19]-code), using
(51−18, 51, 3560)-Net in Base 4 — Upper bound on s
There is no (33, 51, 3561)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 5 083302 332580 970426 998552 563860 > 451 [i]